On the existence of directed strongly regular graphs with parameters $(22, 9, 6, 3, 4)$

The paper shows the existence of the family of directed strongly regular graphs with parameters $(22, 9, 6, 3, 4)$. The adjacency matrices of the found digraphs consist of $3\times 3$ circulant blocks. The automorphism group of all the digraphs found is the group $\mathbb{Z}_3$. The structure of the resulting digraphs has been described using the concepts of skeleton and rigging.